Hydrostatic Pressure (P = ρgh)

Stand a column of fluid up and it presses down with its own weight. The pressure at depth $h$ is $P = \rho g h$ — density times gravity times depth — and the remarkable thing about it is what is not in the formula. There is no area, no volume, no mention of the shape of the container. Pressure at a given depth depends only on how far down you are and what the fluid is. A litre of water in a narrow tube 3 m tall produces exactly the same pressure at its base as a swimming pool 3 m deep.

Three metres down in fresh water: $P = 1000 \times 9.80665 \times 3 \approx 29.4$ kPa. The useful number to carry is that every metre of water is about 9.81 kPa, so every 10 m of water adds roughly one atmosphere — which is why divers count depth in atmospheres and why your ears complain at the deep end of a pool. Mercury, at 13 546 kg/m³, does the same job in 760 mm, which is where that famous barometric height comes from.

Simon Stevin worked this out in the 1580s, and Pascal is said to have demonstrated it by fixing a long thin tube into the top of a sealed barrel and bursting the barrel with a few cups of water poured down the tube. Whether or not the barrel story is literally true, the point stands and is still called the hydrostatic paradox. In practice this equation is mostly used as a translator: divide a pressure by $\rho g$ and you get head in metres, which is the language pump curves are written in.

Three things go wrong, and the density one costs the most money. Head in metres is fluid-specific: a pump rated for 30 m of head delivers $1000 \times 9.80665 \times 30 \approx 294$ kPa on water, but on a 1.20 SG brine the same 30 m of head is 353 kPa, and sizing the system as though a metre were a fixed pressure will have you short. Second, $h$ is vertical depth, measured straight down from the free surface — not the length of pipe, not the run along a sloping hose, and not the distance around a bend. A hundred metres of hose lying flat on the ground develops no static head at all. Third, this is gauge pressure, the amount by which the fluid exceeds the atmosphere above it. For absolute pressure, add about 101 kPa; at 3 m depth in a pool the absolute pressure is roughly 131 kPa, not 29.4.